Discretization of Dirac delta functions in level set methods.

*(English)*Zbl 1074.65025The authors analyze the accuracy of regularizations of Dirac delta functions in the context of the level set method. Following a recent work by A. K. Tornberg and B. Engquist [J. Comput Phys. 200, No. 2, 462–488 (2004; Zbl 1115.76392)] showing that the most common technique for the regularization of the delta function in level set methods is inconsistent and may lead to \(O(1)\) errors, the authors present two techniques to construct consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both techniques are based solely on the distance to the singularity and thus are independent of the grid. The first of them is based on a tensor product of regularized one-dimensional delta functions.

In the second technique a variable support of the regularization domain is used. The regularization is constructed. from a one-dimensional regularization that is extended to multi-dimensions. Being the Dirac delta function \(\delta\) previously replaced by a more regular function \(\delta_\varepsilon\), the multidimensional regularized delta function is then defined as \(\delta_\varepsilon(\Gamma, x)= \delta_\varepsilon(d(\Gamma, x))\) where \(d(\Gamma, x)\) stands for the distance function. Convergence analysis, numerical results and some applications to a class of partial differential equations are also given.

In the second technique a variable support of the regularization domain is used. The regularization is constructed. from a one-dimensional regularization that is extended to multi-dimensions. Being the Dirac delta function \(\delta\) previously replaced by a more regular function \(\delta_\varepsilon\), the multidimensional regularized delta function is then defined as \(\delta_\varepsilon(\Gamma, x)= \delta_\varepsilon(d(\Gamma, x))\) where \(d(\Gamma, x)\) stands for the distance function. Convergence analysis, numerical results and some applications to a class of partial differential equations are also given.

Reviewer: Nácere Hayek (La Laguna)

##### MSC:

65D20 | Computation of special functions and constants, construction of tables |

46F10 | Operations with distributions and generalized functions |

##### Keywords:

discretization of Dirac delta functions; level set methods; consistent approximations; regularized delta functions; regularization; convergence analysis; numerical results
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\textit{B. Engquist} et al., J. Comput. Phys. 207, No. 1, 28--51 (2005; Zbl 1074.65025)

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